The Rule of Plato Project
by Karina Starkova Nov.20
The Rule of Plato Project is about following the rules of two ancient philosophers Plato and Pythagoras to generate Pythagorean Triples, which are the lenghts to form a right triangle. The rules of these two philosophers are somewhat different how to generate Pythagorean Triples, but in this project each rule will be used separately to generate ten different triples, of which five triplet`s calculations will be shown. Then the different methods will be compared to look at how they are alike and different, if there was a pattern and the answer to why they work. Finally all the analyses and discoveries in the project will be summed up and proven.
The rule of Plato states that to generate Pythagorean Triples you need to take an even number,m, the three "legs" of a right triangle are given by sqaring m, dividing the result by four, then subtracting one from the answer and squaring m dividing it by four, but then adding one. The rule of Pythagoras however says that to get the same Pythagorean triples, you need to take an odd number,m, the three 'legs' of a right triangle are given by, sqaring m, subtracting one from it, then dividing the result by two and squaring m, adding one and dividing the answer by two.
In the picture with the turquoise triangle, is an example of what a hypotenuse and what a 'leg' is. The 'legs' are the lengths that make the right angle, which is not labeled in this picture, but still exists. The hypotenuse is the length that connects the lengths and makes the lines into a shape and triangle. In the Pythagorean Theorem the triangle is drawn with three squares attached to each of its sides. The area of the squares is the length of that side squared. But that is the Pythagorean Theorem and not the Pythagorean triples, so the next picture will show the calculations done to get 10 sets of Pythagorean triples using each rule.
Formed with the Rule of Plato
1) 10,24,26 2) 12,35,37 3) 4,3,5 4) 16,63,65 5) 18,80,82 6) 50,624,626 7) 54,728,730 8) 68,1155,1157 9) 76,1443,1445 10) 300,22499,2250
Formed with the Rule of Pythagoras
1) 3,4,5 2) 7,24,25 3) 43,924,925 4) 15,112,113 5) 9,40,41 6) 13,84,85 7) 99,4900,4901 8) 91,4140,4141 9) 55,1512,1513 10) 33,544,545
The two different rules have a lot in common. The first similarity is in the formula. Both formulas subtract one somewhere to get the other leg both formulas add one somewhere in their calculations to get the hypotenuse. Another similarity is their difference between the hypotenuse and the second leg. Their difference is much less that the difference with the first leg, because they are generated from the first leg and increase themselves a lot, but since the formulas to get the second leg and the hypotenuse are a bit alike they have a smaller difference then they have with the first leg. The last similarity that was noticeable was also in the difference. When the number of the first leg gets bigger, the difference between the first leg and the hypotenuse and second leg also increases. That again is because of the squaring in the formulas, which increase the second leg and hypotenuse a lot. The rules of Pythagoras and Plato have many things (mostly in the differences between the numbers) in common.
The most important difference of the two different rules how to come up with a Pythagorean triple, is that one method uses an odd number, the other one an even number. This changes all the calculations, because if the triple is supposed to work the odd number can’t just be switched into the wrong formula otherwise the correct formulas won’t work anymore. Another difference in the two methods is that the numbers of the triples are different. In the rule of Plato the difference between the second leg and the hypotenuse is always 2 and in the rule of Pythagoras it is always 1. The Rules of Plato and Pythagoras are not that different, they simply show how to come up with Pythagorean triples using even and odd numbers.
The biggest pattern is again the difference, but there are also other patterns. For example there is a pattern in the formulas. The formulas have the same structure, the same operations. Another pattern are the triples. If the rule of Plato was used to generate them either one number would be even or all three. In the Pythagorean rule all the triples had two odd numbers and one even. Another patterns and a sign that a set of numbers is a Pythagorean triple is to see if the first number is odd or even and depending on that and look at the difference between the second two numbers. If the number is odd and the difference is one then it is a Pythagorean triple as well as when the numbers is even and the difference is two.
Why the Rules Work
The rules of Pythagoras and Plato to generate a Pythagorean triples work, because in order to create a right triangle two lengths must form the right angle and the third one needs to connect these two lengths to make a triangle and shape out of the angle. The length that connects the legths forming the right angle must of course be longer than both of them; because it needs to connect them and that space will be longer than the lengths creating the right angle. Although the longest length must connect the two other lengths, it does not have a very big difference with the medium sized length, because when all the lengths are squared to form squares around the triangle the lengths forming the triangle must still equal the area of the big square. The length has a lot to do with why the formulas the ancient philosophers Pythagoras and Plato discovered work.
In this project Pythagorean triples were analyzed to understand better why they work. Ten sets of numbers were given for each rule the two ancient philosophers explored. Pythagoras' and Plato's rules had little differences and many similarities, because even if they are different ,the topic is still the same. The differences were the calculations and the relations between the triple numbers. The similarities were also the relations between the numbers and the calculations, but there were two similarities in the difference between the numbers. There was only one pattern and one explanation to why the rules work found in this project. The pattern was a sign that a triple is Pythagorean triple or not, by checking how many odd and even numbers and the difernce between them and the explanation to why these rules work was that the hypotenuse has to connect the lengths forming a right triangle, so it will be just a bit more than the medium length. There are many more explorations about these rules and Pythagorean triples made in the world, but this project only talks about analyzing them without going into the depths of it.
THANK YOU for watching my presentation! I hope you enjoyed it!